Decision Analysis Chapter 12

Chapter Topics Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Components of Decision Making Decision Analysis Components of Decision Making A state of nature is an actual event that may occur in the future. A payoff table is a means of organizing a decision situation, presenting the payoffs from different decisions given the various states of nature. Table 12.1 Payoff Table Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis Decision Making Without Probabilities Figure 12.1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Decision Analysis Decision Making without Probabilities Decision-Making Criteria maximax maximin minimax minimax regret Hurwicz equal likelihood Table 12.2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.3 Payoff Table Illustrating a Maximax Decision Decision Making without Probabilities Maximax Criterion In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion. Table 12.3 Payoff Table Illustrating a Maximax Decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.4 Payoff Table Illustrating a Maximin Decision Decision Making without Probabilities Maximin Criterion In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion. Table 12.4 Payoff Table Illustrating a Maximin Decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.6 Regret Table Illustrating the Minimax Regret Decision Decision Making without Probabilities Minimax Regret Criterion Regret is the difference between the payoff from the best decision and all other decision payoffs. The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. Table 12.6 Regret Table Illustrating the Minimax Regret Decision Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Hurwicz Criterion The Hurwicz criterion is a compromise between the maximax and maximin criterion. A coefficient of optimism, , is a measure of the decision maker’s optimism. The Hurwicz criterion multiplies the best payoff by  and the worst payoff by 1- ., for each decision, and the best result is selected. Decision Values Apartment building $50,000(.4) + 30,000(.6) = 38,000 Office building $100,000(.4) - 40,000(.6) = 16,000 Warehouse $30,000(.4) + 10,000(.6) = 18,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Equal Likelihood Criterion The equal likelihood ( or Laplace) criterion multiplies the decision payoff for each state of nature by an equal weight, thus assuming that the states of nature are equally likely to occur. Decision Values Apartment building $50,000(.5) + 30,000(.5) = 40,000 Office building $100,000(.5) - 40,000(.5) = 30,000 Warehouse $30,000(.5) + 10,000(.5) = 20,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Summary of Criteria Results A dominant decision is one that has a better payoff than another decision under each state of nature. The appropriate criterion is dependent on the “risk” personality and philosophy of the decision maker. Criterion Decision (Purchase) Maximax Office building Maximin Apartment building Minimax regret Apartment building Hurwicz Apartment building Equal likelihood Apartment building Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Solution with QM for Windows (1 of 3) Exhibit 12.1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Solution with QM for Windows (2 of 3) Exhibit 12.2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Solution with QM for Windows (3 of 3) Exhibit 12.3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making without Probabilities Solution with Excel Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 12.4

Decision Making with Probabilities Expected Value Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence. EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000 EV(Office) = $100,000(.6) - 40,000(.4) = 44,000 EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000 Table 12.7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Expected Opportunity Loss The expected opportunity loss is the expected value of the regret for each decision. The expected value and expected opportunity loss criterion result in the same decision. EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000 EOL(Office) = $0(.6) + 70,000(.4) = 28,000 EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000 Table 12.8 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Expected Value Problems Solution with QM for Windows Exhibit 12.5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Expected Value Problems Solution with Excel and Excel QM (1 of 2) Exhibit 12.6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Expected Value Problems Solution with Excel and Excel QM (2 of 2) Exhibit 12.7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Expected Value of Perfect Information The expected value of perfect information (EVPI) is the maximum amount a decision maker would pay for additional information. EVPI equals the expected value given perfect information minus the expected value without perfect information. EVPI equals the expected opportunity loss (EOL) for the best decision. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.9 Payoff Table with Decisions, Given Perfect Information Decision Making with Probabilities EVPI Example (1 of 2) Table 12.9 Payoff Table with Decisions, Given Perfect Information Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities EVPI Example (2 of 2) Decision with perfect information: $100,000(.60) + 30,000(.40) = $72,000 Decision without perfect information: EV(office) = $100,000(.60) - 40,000(.40) = $44,000 EVPI = $72,000 - 44,000 = $28,000 EOL(office) = $0(.60) + 70,000(.4) = $28,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities EVPI with QM for Windows Exhibit 12.8 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.10 Payoff Table for Real Estate Investment Example Decision Making with Probabilities Decision Trees (1 of 4) A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches). Table 12.10 Payoff Table for Real Estate Investment Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Figure 12.2 Decision Tree for Real Estate Investment Example Decision Making with Probabilities Decision Trees (2 of 4) Figure 12.2 Decision Tree for Real Estate Investment Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees (3 of 4) The expected value is computed at each probability node: EV(node 2) = .60($50,000) + .40(30,000) = $42,000 EV(node 3) = .60($100,000) + .40(-40,000) = $44,000 EV(node 4) = .60($30,000) + .40(10,000) = $22,000 Branches with the greatest expected value are selected. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Figure 12.3 Decision Tree with Expected Value at Probability Nodes Decision Making with Probabilities Decision Trees (4 of 4) Figure 12.3 Decision Tree with Expected Value at Probability Nodes Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees with QM for Windows Exhibit 12.9 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees with Excel and TreePlan (1 of 4) Exhibit 12.10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees with Excel and TreePlan (2 of 4) Exhibit 12.11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees with Excel and TreePlan (3 of 4) Exhibit 12.12 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Decision Trees with Excel and TreePlan (4 of 4) Exhibit 12.13 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Sequential Decision Trees (1 of 4) A sequential decision tree is used to illustrate a situation requiring a series of decisions. Used where a payoff table, limited to a single decision, cannot be used. Real estate investment example modified to encompass a ten-year period in which several decisions must be made: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Sequential Decision Trees (2 of 4) Figure 12.4 Sequential Decision Tree Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Sequential Decision Trees (3 of 4) Decision is to purchase land; highest net expected value ($1,160,000). Payoff of the decision is $1,160,000. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Making with Probabilities Sequential Decision Trees (4 of 4) Figure 12.5 Sequential Decision Tree with Nodal Expected Values Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Sequential Decision Tree Analysis Solution with QM for Windows Exhibit 12.14 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Sequential Decision Tree Analysis Solution with Excel and TreePlan Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 12.15

Decision Analysis with Additional Information Bayesian Analysis (1 of 3) Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event. In real estate investment example, using expected value criterion, best decision was to purchase office building with expected value of $444,000, and EVPI of $28,000. Table 12.11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Bayesian Analysis (2 of 3) A conditional probability is the probability that an event will occur given that another event has already occurred. Economic analyst provides additional information for real estate investment decision, forming conditional probabilities: g = good economic conditions p = poor economic conditions P = positive economic report N = negative economic report P(Pg) = .80 P(NG) = .20 P(Pp) = .10 P(Np) = .90 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Bayesian Analysis (3 of 3) A posterior probability is the altered marginal probability of an event based on additional information. Prior probabilities for good or poor economic conditions in real estate decision: P(g) = .60; P(p) = .40 Posterior probabilities by Bayes’ rule: (gP) = P(PG)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923 Posterior (revised) probabilities for decision: P(gN) = .250 P(pP) = .077 P(pN) = .750 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (1 of 4) Decision tree with posterior probabilities differ from earlier versions in that: Two new branches at beginning of tree represent report outcomes. Probabilities of each state of nature are posterior probabilities from Bayes’ rule. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Figure 12.6 Decision Tree with Posterior Probabilities Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (2 of 4) Figure 12.6 Decision Tree with Posterior Probabilities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (3 of 4) EV (apartment building) = $50,000(.923) + 30,000(.077) = $48,460 EV (strategy) = $89,220(.52) + 35,000(.48) = $63,194 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Decision Trees with Posterior Probabilities (4 of 4) Figure 12.7 Decision Tree Analysis Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Computing Posterior Probabilities with Tables Table 12.12 Computation of Posterior Probabilities Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Computing Posterior Probabilities with Excel Exhibit 12.16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Expected Value of Sample Information The expected value of sample information (EVSI) is the difference between the expected value with and without information: For example problem, EVSI = $63,194 - 44,000 = $19,194 The efficiency of sample information is the ratio of the expected value of sample information to the expected value of perfect information: efficiency = EVSI /EVPI = $19,194/ 28,000 = .68 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Utility (1 of 2) Table 12.13 Payoff Table for Auto Insurance Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis with Additional Information Utility (2 of 2) Expected Cost (insurance) = .992($500) + .008(500) = $500 Expected Cost (no insurance) = .992($0) + .008(10,000) = $80 Decision should be do not purchase insurance, but people almost always do purchase insurance. Utility is a measure of personal satisfaction derived from money. Utiles are units of subjective measures of utility. Risk averters forgo a high expected value to avoid a low-probability disaster. Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (1 of 9) Decision Analysis Example Problem Solution (1 of 9) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (2 of 9) Decision Analysis Example Problem Solution (2 of 9) Determine the best decision without probabilities using the 5 criteria of the chapter. Determine best decision with probabilities assuming .70 probability of good conditions, .30 of poor conditions. Use expected value and expected opportunity loss criteria. Compute expected value of perfect information. Develop a decision tree with expected value at the nodes. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20, P(Np) = .80, determine posterior probabilities using Bayes’ rule. Perform a decision tree analysis using the posterior probability obtained in part e. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (3 of 9) Decision Analysis Example Problem Solution (3 of 9) Step 1 (part a): Determine decisions without probabilities. Maximax Decision: Maintain status quo Decisions Maximum Payoffs Expand $800,000 Status quo 1,300,000 (maximum) Sell 320,000 Maximin Decision: Expand Decisions Minimum Payoffs Expand $500,000 (maximum) Status quo -150,000 Sell 320,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (4 of 9) Decision Analysis Example Problem Solution (4 of 9) Minimax Regret Decision: Expand Decisions Maximum Regrets Expand $500,000 (minimum) Status quo 650,000 Sell 980,000 Hurwicz ( = .3) Decision: Expand Expand $800,000(.3) + 500,000(.7) = $590,000 Status quo $1,300,000(.3) - 150,000(.7) = $285,000 Sell $320,000(.3) + 320,000(.7) = $320,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (5 of 9) Decision Analysis Example Problem Solution (5 of 9) Equal Likelihood Decision: Expand Expand $800,000(.5) + 500,000(.5) = $650,000 Status quo $1,300,000(.5) - 150,000(.5) = $575,000 Sell $320,000(.5) + 320,000(.5) = $320,000 Step 2 (part b): Determine Decisions with EV and EOL. Expected value decision: Maintain status quo Expand $800,000(.7) + 500,000(.3) = $710,000 Status quo $1,300,000(.7) - 150,000(.3) = $865,000 Sell $320,000(.7) + 320,000(.3) = $320,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (6 of 9) Decision Analysis Example Problem Solution (6 of 9) Expected opportunity loss decision: Maintain status quo Expand $500,000(.7) + 0(.3) = $350,000 Status quo 0(.7) + 650,000(.3) = $195,000 Sell $980,000(.7) + 180,000(.3) = $740,000 Step 3 (part c): Compute EVPI. EV given perfect information = 1,300,000(.7) + 500,000(.3) = $1,060,000 EV without perfect information = $1,300,000(.7) - 150,000(.3) = $865,000 EVPI = $1.060,000 - 865,000 = $195,000 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (7 of 9) Decision Analysis Example Problem Solution (7 of 9) Step 4 (part d): Develop a decision tree. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (8 of 9) Decision Analysis Example Problem Solution (8 of 9) Step 5 (part e): Determine posterior probabilities. P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)] = (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891 P(pP) = .109 P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)] = (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467 P(pN) = .533 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution (9 of 9) Decision Analysis Example Problem Solution (9 of 9) Step 6 (part f): Decision tree analysis. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall